Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14)(15,24)(16,21)(17,23)(18,32)(19,22)(20,26)(25,41)(27,36)(28,44)(29,39)(30,38)(31,40)(33,35)(34,43)(37,42);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,28)(17,26)(18,33)(19,27)(21,37)(22,35)(23,36)(24,32)(25,30)(29,43)(31,42)(34,38)(39,44)(40,41);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,34)(22,38)(23,40)(24,41)(26,43)(27,28)(32,39)(33,42)(35,37)(36,44);(5,7,8,11)(6,10,9,12)(14,22)(20,35)(21,38)(23,32)(24,36)(34,37)(39,40)(41,44);(13,17,19,18)(14,23,22,32)(15,26,27,33)(16,29,30,31)(20,36,35,24)(21,39,38,40)(25,42,28,43)(34,41,37,44)Polycyclic presentation:
2^9|d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342227);